---
title: "Numerical: Hata path loss for medium city"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:hata-medium-city-pathloss"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/hata-medium-city-pathloss
site: BCT Engineering
---

# Numerical: Hata path loss for medium city

Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks

- Asked in: 2079 Bhadra, 4+4 marks
- Video: [Hata Model Numericals + Formula | Most Asked Exam Topic | Cellular Network](https://www.youtube.com/watch?v=vG6b1Snu-6A) — Vijaya Academy 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/hata-medium-city-pathloss
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

In mobile propagation in a cellular system, find the correction factor and pathloss for a medium size city assuming carrier frequency as 950 MHz, height of transmitting antenna at base station is 45 m, propagation distance between antennas is 10 km and height of receiving antenna in mobile station is 5 m. Compute free space pathloss and compare it with Hata pathloss.

## Short answer

For fc=950 MHz, hte=45 m, hre=5 m, d=10 km, medium-city correction is a(hre)=9.03 dB. Hata urban path loss is 149.64 dB and free-space path loss is 112.0 dB, so Hata is about 37.6 dB higher because urban propagation has extra diffraction, scattering and a larger distance exponent.

## Answer

**Hata model (urban):**

```text
L50 = 69.55 + 26.16 log fc − 13.82 log hte − a(hre)
      + (44.9 − 6.55 log hte) log d
Medium city:
a(hre) = (1.1 log fc − 0.7)hre − (1.56 log fc − 0.8) dB
```
Given: fc = 950 MHz, hte = 45 m, hre = 5 m, d = 10 km. All values are inside Hata's validity range (150–1500 MHz, 30–200 m, 1–10 m, 1–20 km).

#### Correction factor
```text
log 950 = 2.9777
a(hre) = (1.1×2.9777 − 0.7)×5 − (1.56×2.9777 − 0.8)
       = (2.5755)×5 − 3.8452
       = 12.877 − 3.845
       = 9.03 dB
```

#### Hata path loss
```text
26.16 log 950 = 77.90
13.82 log 45  = 22.85
44.9 − 6.55 log 45 = 44.9 − 10.83 = 34.07
log 10 = 1

L50 = 69.55 + 77.90 − 22.85 − 9.03 + 34.07×1
    = 149.64 dB
```

#### Free space path loss
```text
LFS = 32.44 + 20 log f(MHz) + 20 log d(km)
    = 32.44 + 59.55 + 20
    = 112.00 dB
```

#### Comparison
```text
L50 − LFS = 149.64 − 112.00 = 37.64 dB
```

| Quantity | Value |
|---|---|
| Correction factor a(hre) | 9.03 dB |
| Hata path loss | 149.64 dB |
| Free space loss | 112.00 dB |
| Excess loss | 37.64 dB |

**Answer: a(hre) = 9.03 dB, Hata path loss = 149.64 dB, free space loss = 112.0 dB.** The Hata loss is about 37.6 dB higher, because a real city has buildings, ground reflection, diffraction and scattering that free space ignores; also loss in Hata rises at (44.9 − 6.55 log hte) ≈ 34 dB/decade, compared with 20 dB/decade in free space.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which formula is the mobile antenna correction factor for a medium-size city?
   A. a(hre) = (1.1 log fc − 0.7)hre − (1.56 log fc − 0.8)
   B. a(hre) = 3.2 (log 11.75 hre)^2 − 4.97
   C. a(hre) = 20 log(hre/3)
   D. a(hre) = 8.29 (log 1.54 hre)^2 − 1.1
   Answer: A. The medium-city formula uses the (1.1 log fc − 0.7) and (1.56 log fc − 0.8) terms. (Key point: Correction factor formula)

2. What is a(hre) for 950 MHz and hre = 5 m?
   A. 3.85 dB
   B. 12.88 dB
   C. 16.7 dB
   D. 9.03 dB
   Answer: D. (2.5755 x 5) − 3.8452 = 12.877 − 3.845 = 9.03 dB. (Key point: Value of correction factor)

3. What Hata path loss results at d = 10 km?
   A. 112.0 dB
   B. 149.64 dB
   C. 130.2 dB
   D. 186.0 dB
   Answer: B. L50 = 69.55 + 77.90 − 22.85 − 9.03 + 34.07 = 149.64 dB. (Key point: Hata path loss result)

4. Why is the Hata loss about 37.6 dB higher than free-space loss (112 dB)?
   A. Hata ignores antenna heights
   B. Free space includes ground reflection and diffraction
   C. A real city has buildings, reflection, diffraction and scattering that free space ignores
   D. The carrier frequency differs between the two
   Answer: C. Free space assumes a clear path; Hata includes urban obstruction effects, with slope about 34 dB/decade vs 20. (Key point: Comparison with free space)

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